You have 3 equations involving 4 "unknown" parameters. I recommend applying T to (1,0,0,0), (0,1,0,0), (0,0,1,0), and (0,0,0,1) in turn. That will give you 4 vectors in R^3 which clearly cannot be independent. A basis will be a subset of that set of 4 vectors. What space do they span?